9.5 The Binomial Theorem

A binomial is a polynomial that has two terms. The binomial theorem gives you a mechanical way to expand a binomial raised to a certain power.

You can expand

to several powers of n, and make several observations for patterns to help you remember the theorem and understand it better. Let’s look at this binomial raised to several different powers:

· In each binomial expansion, there are (n+1) terms. So if you are raising a binomial to the 7^th power, there will be 8 terms when fully expanded.
· In each expansion, the x power begins with the power the binomial is raised to and descends in each term until it is raised to the 0^th power. The y power begins with 0 and ascends each term until it is raised to the power the whole binomial is being raised to.
· The sum of the powers in each term is n. So no matter what term you are looking at in the expanded form, the exponents of x and y added together will be the number the whole binomial is being raised to.
· The coefficients of each term increase and decrease symmetrically. If you look at the 5^th row, the coefficients are “1, 5, 10, 10, 5, 1”. Whether you start at the beginning or end, the coefficients will be in the same order and be symmetrical.

The Binomial Theorem

-In the expansion of

the coefficient of

**** The symbol

is often used in place of

, to denote binomial coefficients.

Pascal’s Triangle and Its Patterns

The first and last number in each row is 1. Every other number in each row is found by adding the two numbers immediately above the number. The numbers in this triangle are exactly the same numbers as the coefficients of binomial expansions.